Quantum theory of magnetic octupole in periodic crystals and characterization of time-reversal-symmetry breaking antiferromagnetism

Magnetic multipoles have been recognized as order parameters characterizing magnetic structure in solids, with magnetic dipole serving as canonical examples in ferromagnets. Recently, magnetic octupoles have been proposed as the order parameters of time-reversal-symmetry breaking centrosymmetric antiferromagnets exhibiting nonrelativistic spin splitting, which is referred to as ``altermagnet''. However, a gauge-invariant formulation of magnetic octupoles in crystalline solids remains elusive. Here, we present a gauge-invariant expression of spin magnetic octupoles in periodic crystals based on quantum mechanics and thermodynamics. Our expression reveals a contribution from an anisotropic magnetic dipole, which has the same symmetry as conventional spin and orbital magnetic dipoles but carries no net magnetization. We demonstrate that such an anisotropic magnetic dipole acts as an order parameter for antiferromagnets that exhibit the anomalous Hall effect.